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Abstract

Distributed manipulation systems induce motions on objects through the application of many external forces. An actuator array performs distributed manipulation using a planar array of many small stationary elements (which are called cells) that cooperate to manipulate larger objects. Typically, highly dense actuator arrays are modeled as spatially continuous, programmable force fields, although in many implementations a relatively small number of actuators supports an object and continuous assumptions break down. This paper serves two purposes: to present a methodology for modeling and analyzing the dynamics of manipulation on a highly discrete actuator array and to present a methodology for designing manipulation strategies on discrete actuator arrays. This is done in the context of a particular macro-scale actuator array comprising a fixed planar array of motorized wheels. Modeling of the dynamics takes into account several models of the interaction between the actuators and the object, the distribution of the weight of the object among the supports, and the discrete nature of the system. Under certain modeling assumptions, the manipulation dynamics of an object are extremely simple for a given set of supporting cells. An inversion of these piecewise-continuous dynamics generates a fully continuous open-loop manipulation strategy, effectively smoothing out the discontinuities. The authors show that although the resulting manipulation field may stably position and orient any object in the continuous field case, discreteness causes many objects to experience unstable rotational equilibria. Thus, poor orientation precision is a limitation of open-loop manipulation using discrete actuator arrays and motivates the use of feedback. The authors also derive closed-loop manipulation strategies through an inversion of the discrete dynamics that reduce the many-input, three-output distributed control problem to a standard three-input, three-output control problem that operates under distributed control. In effect, the array of actuators is reduced to a single virtual actuator capable of applying a desired net force and moment on an object. It is proven that even in the presence of dynamic coupling and nonlinearities introduced due to discreteness, these closed-loop strategies are asymptotically stable. Multimedia extensions include a complete simulator and videos of the experimental prototype.